IV-1 



IV. WEAK INHOMOGENEITIES 



The properties of the ocean medium, as encountered by a sound wave 

 passing through it, are constantly varying even in the absence of the strong 

 inhomogeneities discussed in the previous chapter. The pressure, salinity, 

 and temperature exhibit small variations both in the horizontal plane and in the 

 vertical direction. The variations in the vertical direction tend to exhibit a 

 layer structure, whereas the horizontal variations are more in the nature of 

 random patches . In this chapter we shall examine the effect of these inhomoge- 

 neities on the propagation of sound waves. 



A. THE WAVE EQUATION FOR AN INHOMOGENEOUS MEDIUM 



The derivation of the acoustic wave equation appropriate for an ocean 

 medium with slightly varying properties has been dealt with in considerable detail 

 by previous authors, including a report in the present series. ^ The variations in 

 pressure, salinity, temperature, and gravitational field introduce a number of 

 corresponding additional terms beyond those appearing in the ordinary wave 

 equation for a homogeneous medium. An analysis of the order of magnitude of the 

 different additional terms under actual ocean conditions indicates that the dominant 

 effect for the analysis of sound propagation comes from the spatial variations of 

 the velocity of sound c caused by the variations in temperature and salinity. The 

 magnitude of this effect exceeds the effects of spatial variations in density by at 

 least a factor of 10. For our purposes, therefore, it will be adequate to confine 

 ourselves to a wave equation with a slightly varying velocity of sound: 



? 1^ - v = p (IV-l) 



The velocity of sound, c , is a slightly varying function of position and time. Let 

 Cq be a typical average value of the sound velocity in the spatial region in which we 

 wish to study acoustic wave propagation. We may then introduce the local index of 

 refraction of the medium, n , as the ratio of the average sound velocity to the 

 local sound velocity: 



c 

 n(X' t) = c(x°t) = 1 + u (x, t) (IV-2) 



1. Arthur D. Little, Inc., Ref. IV-4, Part A, Section 1-14 

 Chernov, Ref. IV-2, Part II, Chapters III and IV 



:arthur Hi.liittU.llnc. 



S-7001-Q307 



